To implement such an unsupervised transcription system, we need a synthesis system, \(\hat{x}=F_s(\hat{y})\) , making the overall system \(\hat{x}=F_s(F_a(x))\) . During its training, the system is given \(\lbrace x\rbrace _{m=1}^M\)  and trained to minimize the distance between \(x\)  and \(\hat{x}\) .
There have been few systems relying on unsupervised learning as explained above.
In MIR, the system in [1]} utilized sparse coding to learn a dictionary of time-frequency templates of piano and harpsicord, assuming a (matrix-)multiplication model with additive noise, \(F_s(\mathbf {y})=\mathbf {Ay}+\mathbf {e}\) . Yoshii et al. proposed to use sparse coding in a jointly-learned chord recognition and transcription system  [2]}. Berg et al. designed a probabilistic graphical model that parameterizes the spectral and temporal envelopes, note events, and note activations, in order to transcribe piano by inferring their parameters  [3]}.
In drum transcription, many systems have used NMF to decompose a drum track spectrum into spectral templates and their temporal activations (or transcription) [4]}, [5]}. Several variants of NMF were proposed to address the limits of the fixed spectrum template of NMF [6]}, [7]}, [8]}.
Lastly, a similar system can be found in computer vision, where the parameters of input images are estimated by reconstruction using an image renderer [9]}.
